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218
KazhdanLusztig polynomials for 321hexagonavoiding permutations
 J. ALG. COMB
, 2001
"... In (Deodhar, Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the KazhdanLusztig polynomials Px,w in the case where W is any Coxeter group. We explicitly describe the combinatorics in the case where W = Sn (the symmetric group on n letters) and the ..."
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Cited by 50 (5 self)
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In (Deodhar, Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the KazhdanLusztig polynomials Px,w in the case where W is any Coxeter group. We explicitly describe the combinatorics in the case where W = Sn (the symmetric group on n letters) and the permutation w is 321hexagonavoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for w. As a consequence of our results on KazhdanLusztig polynomials, we show that the Poincaré polynomial of the intersection cohomology of the Schubert variety corresponding to w is (1 + q) l(w) if and only if w is 321hexagonavoiding. We also give a sufficient condition for the Schubert variety Xw to have a small resolution. We conclude with a simple method for completely determining the singular locus of Xw when w is 321hexagonavoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (Bn, F4, G2).
Cohomological aspects of Magnus expansions
, 2006
"... We generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group of a free group Aut(Fn). The extended ones ar ..."
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Cited by 48 (7 self)
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We generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group of a free group Aut(Fn). The extended ones are not homomorphisms, but satisfy an infinite sequence of coboundary relations, so that we call them the Johnson maps. In this paper we confine ourselves to studying the first and the second relations, which have cohomological consequences about the group Aut(Fn) and the mapping class groups for surfaces. The first one means that the first Johnson map is a twisted 1cocycle of the group Aut(Fn). Its cohomology class coincides with “the unique elementary particle ” of all the MoritaMumford classes on the mapping class group for a surface [Ka1] [KM1]. The second one restricted to the mapping class group is equal to a fundamental relation among twisted MoritaMumford classes proposed by Garoufalidis and Nakamura [GN] and established by Morita and the author [KM2]. This means we give a simple and coherent proof of the fundamental relation. The first Johnson map gives the abelianization of the induced automorphism group IAn of a free group in an explicit way.
Schubert Polynomials, The Bruhat Order, And The Geometry Of Flag Manifolds
 Duke Math. J
, 1998
"... We illuminate the relation between the Bruhat order and structure constants for the polynomial ring in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We est ..."
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Cited by 47 (21 self)
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We illuminate the relation between the Bruhat order and structure constants for the polynomial ring in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We establish a number of new identities among these structure constants. This leads to formulas for some of these constants and new results on the enumeration of chains in the Bruhat order. A new graded partial order on the symmetric group which contains Young's lattice arises from these investigations. We also derive formulas for certain specializations of Schubert polynomials.
KostkaFoulkes polynomials and Macdonald spherical functions
 in Surveys in Combinatorics 2003, C. Wensley ed., London Math. Soc. Lect. Notes 307 Camb
, 2003
"... Abstract. Generalized HallLittlewood polynomials (Macdonald spherical functions) and generalized KostkaFoulkes polynomials (qweight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different defin ..."
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Cited by 34 (4 self)
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Abstract. Generalized HallLittlewood polynomials (Macdonald spherical functions) and generalized KostkaFoulkes polynomials (qweight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from “scratch”, in a presentation which, hopefully, will be accessible and useful for both the nonexpert and researchers currently working in this very active field. The combinatorics of the affine Hecke algebra plays a central role. The final section of this paper can be read independently of the rest of the paper. It presents, with proof, Lascoux and Schützenberger’s positive formula for the KostkaFoulkes poynomials in the type A case. The classical theory of HallLittlewood polynomials and the KostkaFoulkes polynomials appears in the monograph of I.G. Macdonald [Mac]. The HallLittlewood polynomials form a basis of the ring of symmetric functions and the KostkaFoulkes polynomials are the entries of the transition matrix between the HallLittlewood polynomials and the Schur functions.
Extended affine Lie algebras
"... Abstract. In this announcement we describe the structure of an extended affine Lie algebra in terms of its centreless core. 0. Introduction. Extended affine Lie algebras are a class of complex Lie algebras that includes finitedimensional simple Lie algebras, affine Lie algebras and toroidal Lie alg ..."
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Cited by 34 (1 self)
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Abstract. In this announcement we describe the structure of an extended affine Lie algebra in terms of its centreless core. 0. Introduction. Extended affine Lie algebras are a class of complex Lie algebras that includes finitedimensional simple Lie algebras, affine Lie algebras and toroidal Lie algebras. They are closely related to Saito’s elliptic Lie algebras ([29]). Originally proposed by the physicists HøeghKrohn and B. Torrésani [20] under the name irreducible quasisimple Lie algebras, extended affine Lie algebras have been put on a sound mathematical footing in the AMSmemoirs [2] by Allison,
The Rost invariant has trivial kernel for quasisplit groups of low rank
 Comment. Math. Helv
"... Abstract. For G an almost simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map RG: H 1 (F, G) → H 3 (F, Q/Z(2)). This includes the Arason invariant for quadratic forms and Rost’s mod 3 invariant for Albert algebras as special cases. We sho ..."
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Cited by 33 (3 self)
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Abstract. For G an almost simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map RG: H 1 (F, G) → H 3 (F, Q/Z(2)). This includes the Arason invariant for quadratic forms and Rost’s mod 3 invariant for Albert algebras as special cases. We show that RG has trivial kernel if G is quasisplit of type E6 or E7. A casebycase analysis shows that it has trivial kernel whenever G is quasisplit of low rank. For G an almost simple simply connected algebraic group over a field F, the set of all natural transformations of functors H 1 (?, G) − → H 3 (?, Q/Z(2)) is a finite cyclic group [KMRT98, §31] with a canonical generator. (Here Hi (?, M) is the Galois cohomology functor which takes a field extension of your base field F and returns a group if M is abelian and a pointed set otherwise. When F has characteristic zero, Q/Z(2) is defined to be lim → µ ⊗2 n for µ n the algebraic groups of nth roots of unity; see [KMRT98, p. 431] or [Gilb, I.1(b)] for a more complete definition.) This generator is called the Rost
Beyond KirillovReshetikhin modules
"... we shall be concerned with the category of finite–dimensional representations of the untwisted quantum affine algebra when the quantum parameter q is not a root of unity. We review the foundational results of the subject, including the Drinfeld presentation, the classification of simple modules and ..."
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Cited by 33 (12 self)
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we shall be concerned with the category of finite–dimensional representations of the untwisted quantum affine algebra when the quantum parameter q is not a root of unity. We review the foundational results of the subject, including the Drinfeld presentation, the classification of simple modules and qcharacters. We then concentrate on particular families of irreducible representations whose structure has recently been understood:
Category O: Quivers and Endomorphism Rings of Projectives
, 2003
"... We describe an algorithm for computing quivers of category O of a finite dimensional semisimple Lie algebra. The main tool for this is Soergel’s description of the endomorphism ring of the antidominant indecomposable projective module of a regular block as an algebra of coinvariants. We give explic ..."
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Cited by 32 (9 self)
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We describe an algorithm for computing quivers of category O of a finite dimensional semisimple Lie algebra. The main tool for this is Soergel’s description of the endomorphism ring of the antidominant indecomposable projective module of a regular block as an algebra of coinvariants. We give explicit calculations for root systems of rank 1 and 2 for regular and singular blocks and also quivers for regular blocks for type A3. The main result in this paper is a necessary and sufficient condition for an endomorphism ring of an indecomposable projective object of O to be commutative. We give also an explicit formula for the socle of a projective object with a short proof using Soergel’s functor V and finish with a generalization of this functor to HarishChandra bimodules and parabolic versions of category O.
Coxeter groups, Salem numbers and the Hilbert metric
, 2001
"... this paper we prove a similar result for loops in the fundamental polyhedron of a Coxeter group W , and use it to study the spectral radius (w), w 2 W for the geometric action of W . In particular we prove: Theorem 1.1 Let (W; S) be a Coxeter system and let w 2 W . Then either (w) = 1 or (w) Le ..."
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Cited by 32 (6 self)
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this paper we prove a similar result for loops in the fundamental polyhedron of a Coxeter group W , and use it to study the spectral radius (w), w 2 W for the geometric action of W . In particular we prove: Theorem 1.1 Let (W; S) be a Coxeter system and let w 2 W . Then either (w) = 1 or (w) Lehmer 1:1762808. Here Lehmer denotes Lehmer's number, a root of the polynomial 1 + x \Gamma x 3 \Gamma x 4 \Gamma x 5 \Gamma x 6 \Gamma x 7 + x 9 + x 10 (1.1) and the smallest known Salem number. Billiards. Recall that a Coxeter system (W; S) is a group W with a finite generating set S = fs 1 ; : : : ; s n g, subject only to the relations (s i s j ) m ij = 1, where m ii = 1 and m ij 2 for i 6= j. The permuted products s oe1 s oe2 \Delta \Delta \Delta s oen 2 W; oe 2 S n ; are the Coxeter elements of (W; S). We say w 2 W is essential if it is not conjugate into any subgroup W I ae W generated by a proper subset I ae S. The Coxeter group W acts naturally by reflections on V